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A computer simulation study of the effect of pressure on Mg diffusion in forsterite
Frédéric B’Ejina, Marc Blanchard, Kate Wright, G. David Price
To cite this version:
Frédéric B’Ejina, Marc Blanchard, Kate Wright, G. David Price. A computer simulation study of the effect of pressure on Mg diffusion in forsterite. Physics of the Earth and Planetary Interiors, Elsevier, 2008, 172 (1-2), pp.13. �10.1016/j.pepi.2008.04.008�. �hal-00532144�
Accepted Manuscript
Title: A computer simulation study of the effect of pressure on Mg diffusion in forsterite
Authors: Fr´ed´eric B’ejina, Marc Blanchard, Kate Wright, G.
David Price
PII: S0031-9201(08)00077-0
DOI: doi:10.1016/j.pepi.2008.04.008 Reference: PEPI 4928
To appear in: Physics of the Earth and Planetary Interiors Received date: 12-10-2007
Revised date: 10-4-2008 Accepted date: 18-4-2008
Please cite this article as: B’ejina, F., Blanchard, M., Wright, K., Price, G.D., A computer simulation study of the effect of pressure on Mg diffusion in forsterite, Physics of the Earth and Planetary Interiors (2007), doi:10.1016/j.pepi.2008.04.008
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Accepted Manuscript
A computer simulation study of the effect of pressure on Mg diffusion in forsterite
Fr´ ed´ eric B´ ejina
a,∗
aLaboratoire des M´ecanismes et Transferts en G´eologie, Universit´e de Toulouse - CNRS - IRD - OMP, 14, avenue ´Edouard Belin, 31400 Toulouse, France
Marc Blanchard
bbInstitut de Min´eralogie et Physique des Milieux Condens´es (IMPMC), UMR CNRS 7590, Universit´e Paris VI, Universit´e Paris VII, IPGP, 140 rue de
Lourmel, 75015 Paris, France
Kate Wright
ccNanochemistry Research Institute, Curtin University of Technology, PO Box U1987, Perth 6845, WA, Australia
G. David Price
ddDepartment of Earth Sciences, UCL, Gower Street, London WC1E 6BT, UK Abstract
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Abstract
Computer simulation techniques were used to investigate the effect of pressure on magnesium diffusion in forsterite between 0 and 10 GPa. We studied the diffusion path along the ~c crystallographic axis (we always refer to the Pbnm system) via a vacancy mechanism. Using a Mott-Littleton approach within the code GULP, we were able to precisely map the diffusion path of a Mg vacancy and we found the activation energy, E = 3.97 eV at 0 GPa (with Ef = 3.35 eV for the formation energy and Em = 0.62 eV for the migration) and E = 4.46 eV at 10 GPa (Ef = 3.81 eV and Em= 0.65 eV). Preliminary results using the supercell technique gave the same saddle point coordinates and energies. This saddle point of the Mg vacancy diffusion found with GULP was then introduced in an ab initio code, confirming the values of the migration energy both at 0 and 10 GPa. We were therefore able to estimate the activation volume (∆V ) to be around 5 cm3/mol and d(∆V )/dP ≃ 0.
The effect of pressure applies mostly on defect formation and little on migration.
Key words: Forsterite, Computational modeling, Point defect, Diffusion, Activation volume
61.72.Ji, 66.30.-h, 82.20.Wt, 91.40.Ac, 91.60.Dc, 91.60.Gf
∗ Corresponding author
Email addresses: bejina@lmtg.obs-mip.fr (Fr´ed´eric B´ejina),
marc.blanchard@impmc.jussieu.fr(Marc Blanchard), kate@ivec.org (Kate Wright), d.price@ucl.ac.uk (G. David Price).
Accepted Manuscript
A computer simulation study of the effect of pressure on Mg diffusion in forsterite
Fr´ed´eric B´ejina
a,∗
aLaboratoire des M´ecanismes et Transferts en G´eologie, Universit´e de Toulouse - CNRS - IRD - OMP, 14, avenue ´Edouard Belin, 31400 Toulouse, France
Marc Blanchard
bbInstitut de Min´eralogie et Physique des Milieux Condens´es (IMPMC), UMR CNRS 7590, Universit´e Paris VI, Universit´e Paris VII, IPGP, 140 rue de
Lourmel, 75015 Paris, France
Kate Wright
ccNanochemistry Research Institute, Curtin University of Technology, PO Box U1987, Perth 6845, WA, Australia
G. David Price
ddDepartment of Earth Sciences, UCL, Gower Street, London WC1E 6BT, UK
* Manuscript
Accepted Manuscript
Abstract
Computer simulation techniques were used to investigate the effect of pressure on magnesium diffusion in forsterite between 0 and 10 GPa. We studied the diffusion path along the ~c crystallographic axis (we always refer to the Pbnm system) via a vacancy mechanism. Using a Mott-Littleton approach within the code GULP, we were able to precisely map the diffusion path of a Mg vacancy and we found the activation energy, E = 3.97 eV at 0 GPa (with Ef = 3.35 eV for the formation energy and Em = 0.62 eV for the migration) and E = 4.46 eV at 10 GPa (Ef = 3.81 eV and Em= 0.65 eV). Preliminary results using the supercell technique gave the same saddle point coordinates and energies. This saddle point of the Mg vacancy diffusion found with GULP was then introduced in an ab initio code, confirming the values of the migration energy both at 0 and 10 GPa. We were therefore able to estimate the activation volume (∆V ) to be around 5 cm3/mol and d(∆V )/dP ≃ 0.
The effect of pressure applies mostly on defect formation and little on migration.
Key words: Forsterite, Computational modeling, Point defect, Diffusion, Activation volume
61.72.Ji, 66.30.-h, 82.20.Wt, 91.40.Ac, 91.60.Dc, 91.60.Gf
1 Introduction
A knowledge and complete understanding of defects and diffusion properties in minerals is of major importance for studying transport mechanisms in the
∗ Corresponding author
Email addresses: bejina@lmtg.obs-mip.fr (Fr´ed´eric B´ejina),
marc.blanchard@impmc.jussieu.fr(Marc Blanchard), kate@ivec.org (Kate Wright), d.price@ucl.ac.uk (G. David Price).
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Earth such as viscosity, electrical conductivity, etc. In addition, understanding the dynamics of the interior of the Earth necessitates the performance of min- eral physics experiments at high pressure, a very challenging task. Theoretical calculations of physical properties of minerals can be of great help, particu- larly for diffusion studies for which they open a window on the microscopic processes involved. They also permit the separation of the contributions of de- fect formation and migration to the activation energy, which is very difficult to do experimentally.
Since the early work by Naughton and Fujikawa (1959), divalent cation dif- fusion in olivines has been extensively studied for forsterite: (Morioka, 1981;
Andersson et al., 1989; Chakraborty et al., 1994; Farver et al., 1994) and refer- ences in (Freer, 1981); and for Fe-bearing olivines: see refs in (Dohmen et al., 2007). Olivine is the only mantle mineral whose defect and diffusion proper- ties, including those at elevated pressures (for review see B´ejina et al., 2003,
; also a recent work by Holzapfel et al. (2007)) are currently well understood.
Several studies have confirmed that divalent cation diffusion is faster along the ~c (Pbnm) axis (Clark and Long, 1971; Buening and Buseck, 1973; Mis- ener, 1974; Jurewicz and Watson, 1988; Andersson et al., 1989). Buening and Buseck (1973) also proposed that diffusion occurs via a vacancy mechanism which has not been refuted since.
Results of computational studies on defects and diffusion in olivine are becom- ing more and more available: for example, on OH defects (Wright and Catlow, 1994; Brodholt and Refson, 2000; Braithwaite et al., 2002, 2003) and on oxy- gen defects and diffusion (Brodholt, 1997; Walker et al., 2003). Note that Si diffusion remains difficult to investigate using atomistic methods as discussed by Braithwaite et al. (2003). Walker et al. (2008) studied Mg diffusion along
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the three crystallographic axes (see also (Brodholt, 1997) for the energetics of Mg point defects), and confirmed that diffusion along ~c presents the lowest activation energy. Mg diffusion was previously studied by Jaoul et al. (1995), but at high pressure, who found an activation volume for the migration of Mg close to 0. We have reexamined this quantity by mapping precisely the saddle point using the RFO procedure within the code GULP (Gale, 1997; Gale and Rohl, 2003) as explained below, a feature not available in the code CASCADE (Leslie, 1981) used by these authors. It was also necessary to compare differ- ent computer simulation techniques in order to verify that a small activation volume couldn’t be attributed to the approach used by Jaoul et al. (1995). Ac- cording to previous experimental and theoretical works, Mg diffusion along ~c may very well be the dominant Me-cation transport mechanism in the Earth’s mantle, and therefore, precise determination of the its activation volume is crucial.
In this paper, we present our results on Mg diffusion along the ~c axis via a vacancy mechanism at high pressure, using two atomistic approaches (Mott- Littleton and supercell) as well as ab initio.
2 Methodology
In the following sections, the term supercell is reserved for the atomistic calcu- lations only. The CASTEP calculations, which also use a supercell, are referred to as ab initio.
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2.1 Atomistic methods
The migration of a Mg vacancy along the ~c axis of forsterite was studied with the Mott-Littleton (ML) approach, and preliminary calculations were performed with the atomistic supercell method. In both cases we used the code GULP (Gale, 1997; Gale and Rohl, 2003) and all calculations employed a static lattice approach, i.e. were done at 0 K.
ML calculations were run at constant pressure with a method identical to Walker et al. (2008) (further details can also be found in Walker et al., 2003).
The supercell method was run at constant cell parameters as in (Richmond and Brodholt, 2000); see also an example and description in (Voˇcadlo et al., 1995). Both ML and supercell calculations were performed with the well-tested set of interatomic potentials (Catlow, 1977; Sanders et al., 1984; Lewis, 1985;
Lewis and Catlow, 1985, ; all parameters are given in Table 1), called THB1 by Price et al. (1987b). In addition the polarization of oxygen was described using the shell model of Dick and Overhauser (1958). These potentials, which were also used by Wright and Catlow (1994) and Jaoul et al. (1995) with the code CASCADE (Leslie, 1981), have proven to be very reliable for diffusion studies.
The ML method is based on dividing the crystal into two concentric regions, region 1 and region 2 (itself divided in regions 2a and 2b; e.g., description in (Gale and Rohl, 2003)). Our forsterite model used a region 1 of 550 ions (radius 11) surrounded by region 2a of radius 24 ˚A. The force acting on region 2a ions is estimated using the electrostatic force of the defects present in region 1. For our preliminary supercell work we used a cell containing 224 ions
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((2 × 2 × 2) Fo unit cells). Because we considered the migration of a Mg2+
ion between two Mg vacancies, the system was charged. With the supercell method, a uniform background charge is added to the cell in order to make it neutral and to cancel the divergence of the electrostatic energy. While in the ML approach one deals with a single charge, the supercell method repeats the charge periodically. Therefore, the total energy has to be corrected for electrostatic defect-defect interactions generated by the periodic repetition of the cell in the three directions. The method, explained in (Leslie and Gillan, 1985), has already been applied to forsterite (e.g., Brodholt, 1997). For a cubic unit cell, this corrective term is:
Ecorr = αq2
2ε0L, (1)
where α is the Madelung constant for the cell, L is the cell length, ε0 is the dielectric constant for the perfect crystal, and q is the net charge of the defect.
Following Brodholt (1997) who used the Ewald Sum technique to calculate Ecorr for an orthorhombic unit cell, we found that Ecorr (eV) = 1.253 q2/ε0
at 0 GPa and 1.294 q2/ε0 at 10 GPa. For the dielectric constant of forsterite, we used the value of 6.2 also chosen by Brodholt (1997). Other values have been proposed (e.g., Cygan and Lasaga, 1986; Shannon and Subramanian, 1989), but would not affect our conclusions. This correction uses an average value for the dielectric constant. Recently, Woodley et al. (2003) proposed a correction based on the trace of the dielectric tensor (see their Eq. (4)) but, because forsterite doesn’t present a strong anisotropy, the approximation of an isotropic dielectric continuum remains valid.
The Mg vacancy (VMg) diffusion was studied by looking at the path that a Mg interstitial (Mgi) follows when jumping from a M1 vacant site into the
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adjacent one along [001]. The choice of M1 vacancies was guided by previous experimental studies as explained above, but also by several theoretical works, including our own (Table 2), that have shown M1 to be more favorable than M2 vacancies (Jaoul et al., 1995; Brodholt, 1997; Richmond and Brodholt, 2000; Walker et al., 2008). Using both ML and supercell, the Mg migration path was first found “manually” at 0 and 10 GPa, as described in Fig. 1.
We also used the Rational Function Optimization procedure (Banerjee et al., 1985) within GULP in order to precisely locate the transition state (or saddle point), after it had approximately been located “manually”. We verified that at 0 and 10 GPa, both manual and RFO localizations gave very close positions of the saddle point. This allowed us to run only the RFO procedure to locate the transition state at intermediate pressures (only with ML).
2.2 Ab initio calculations
The migration energy at 0 and 10 GPa was subsequently checked by first- principles calculations (code CASTEP, Segall et al., 2002). The Mg intersti- tial was fixed at the saddle point found with GULP while all other ions were free to relax. These calculations were performed in an optimized (2 × 1 × 2) supercell. In this density functional theory code, the wavefunctions are ex- panded in plane-waves. Only the valence electrons were considered explic- itly through the use of ultrasoft pseudopotentials (Vanderbilt, 1990). The PW91 gradient-corrected functional was employed for the determination of the exchange-correlation energy (Perdew and Wang, 1992). Considering the size of the system (112 ions), only calculations at the center of the Brillouin zone (Γ-point) with an energy cut-off of 600 eV were computationally afford-
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able. However, these parameters lead to a satisfactory convergence of the total energy. Finally, because the defect-defect interaction energy does not vary with the location of the defect in the simulated structure, it has no effect on the migration energy.
3 Results
The ability of the potential set, THB1, used in our atomistic simulations, to reproduce the structural and elastic properties of forsterite has already been discussed elsewhere (e.g., Price et al., 1987a,b; Richmond and Brodholt, 2000).
The forsterite structure and compressibility are also well described by our ab initio calculations: at 0 GPa, a = 4.792 ˚A, b = 10.322 ˚A and c = 6.054 ˚A (V = 299.45 ˚A3) in good agreement with experimental values (Hazen, 1976); at 10 GPa, a = 4.714 ˚A, b = 10.009 ˚A, c = 5.910 ˚A and V = 278.85 ˚A3 (V /V0 = 0.9312), also in excellent agreement with experiments on compressibility (see, for example, Fig. 2 in Wentzcovitch and Stixrude, 1997).
3.1 Point defects
The calculations of the formation energies of individual defects were carried out using the atomistic ML and supercell methods. As shown in Tables 2 and 3, ML and supercell methods gave very similar values, thus our discussion is limited to the ML results unless mentioned otherwise.
Vacancies are the easiest defects to study because they simply consist of remov- ing one or several ions from the structure. The energies of vacancy formation in forsterite are in excellent agreement with previous works (Tables 2 and 3).
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For Mg vacancies, we found that the M1 site is more favorable than the M2 by about 1.9 eV at 0 GPa and by about 2.2 at 10 GPa. The effect of pressure on the formation energy of VMg is about twice as large for the M2 site compared to M1 (∆Ef(M2) = 0.52 eV and ∆Ef(M1) = 0.24 eV; see also Fig. 2) sug- gesting a slightly different compressibility between the two sites (Jaoul et al., 1995; Brodholt et al., 1996; Wentzcovitch and Stixrude, 1997).
Interstitial defects present the difficulty of locating the proper site in the forsterite structure. Walker et al. (2008) did a systematic search to locate the most stable configuration for Mgi. They found the lowest formation energy for a split interstitial across a M1 site (see their Fig. 3). This favorable split interstitial was previously found by Jaoul et al. (1995) and both our ML and supercell calculations reproduced this defect, with a comparable formation energy (Tables 2 and 3). In Table 3, one can see that supercell calculations gave slightly lower energies (by about 0.2 eV) at both 0 and 10 GPa for the formation of Mgi whereas they were quasi-identical for vacancies. This small difference can be explained either by the fact that the split-interstital defect may carry a significant multipole in each of the repeating supercells and whose interactions could lower the energy or, by considering the effect of interstitial and vacancy on the cell volume. The supercell calculations were performed at constant volume and thus the cell parameters were unable to adjust to the perturbation caused by the interstitial. The effect of pressure therefore varies also between the two methods.
Because they are charged species, vacancies and interstitials must occur as part of a neutral Schottky or Frenkel defect. The Mg-Frenkel pair (VMg(M1) + Mgi) has been reported as being the most favorable, both at 0 and 10 GPa (Jaoul et al., 1995; Walker et al., 2008). From Tables 2 and 3, we found that Ef =
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3.35 eV at 0 GPa and 3.81 eV at 10 GPa, in perfect agreement with these previous studies. The activation volume for the formation of the Mg-Frenkel pair calculated between 0 and 10 GPa is therefore,
∆Vf = EfFrenkel(10 GPa) − EfFrenkel(0 GPa)
∆P = 4.43 cm3/mol, (2)
with EfFrenkel the formation energy of the Frenkel pair of defects and ∆P the pressure range. Fig. 3 shows the formation energy obtained at intermediate pressures. ∆Vf can be considered constant over the investigated pressure range although one can see a slight curvature denoting a small decrease of ∆Vf as pressure increases.
3.2 Migration of VMg along [001]
The diffusion path between two adjacent M1 vacant sites along [001] was found to be identical at 0 and 10 GPa (Fig. 4 with its energy profile depicted in Fig. 5). At mid distance between the two Mg vacancies lies a (001) mirror plane and, therefore, the diffusion path should be symmetrical on both sides.
We precisely determined only half of it and checked a few points on the other side of the mirror finding that the symmetry was indeed respected. As one can expect from the crystal structure, we found that the energy minimum of diffusion when Mgi is in an empty tetrahedron-like site on the (001) mirror plane, and that Mgi leaves (and enters) a M1 site through the middle of the face of the octahedron (Fig. 4). The migration energy of diffusion is defined at the saddle point, the position for the highest energy along the diffusion path, i.e. the energy barrier that Mgihas to cross. This position was found at (0.53, 0.45, 0.32) and (0.53, 0.44, 0.31) at 0 and 10 GPa respectively and corre-
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sponds to the coordinates when Mgi is leaving the Mg vacancy and is crossing the face of the empty octahedron; the corresponding energy is 25.09 eV at 0 GPa and 25.36 eV at 10 GPa. The migration activation energy is therefore Em = 0.62 eV at 0 GPa and 0.65 eV at 10 GPa, corresponding to an activation volume for migration of ∆Vm= 0.29 cm3/mol, in agreement with Jaoul et al.
(1995) who found ∆Vm = 0 cm3/mol. Note that the supercell method gave the same saddle point coordinates but slightly different energies, by about 0.2 eV, for the same reason as explained above for Mgi. Finally, because the saddle point coordinates at 0 and 10 GPa are very close, we were able to run the RFO calculations at 2.5, 5 and 7.5 GPa without the manual search. It shows that the saddle point is identical at all pressures (Fig. 5) and that pressure has no noticeable effect on ∆Vm between 0 and 10 GPa (Fig. 3).
Finally, we checked the migration energy obtained with GULP (from ML and supercell methods) with ab initio calculations. We used the same coordinates for the saddle point as found with ML. At 0 GPa, the migration energy we found was Em = 0.84 eV, and at 10 GPa, Em = 0.91 eV; the activation volume for the migration is therefore, ∆Vm = 0.67 cm3/mol. These energies are per- fectly compatible with ML results and even more with the atomistic supercell method. Because ab initio calculations are computationally very demanding, we did not search for the saddle point. Its coordinates could very well be slightly different than for GULP calculations and the migration energy lower.
However, we do not believe the migration energy could be lowered by more than 0.1 eV because the energy surface around the saddle point is relatively flat.
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4 Discussion and conclusion
In this section, we only discuss our results on the activation volume. Our acti- vation energy is in perfect agreement withWalker et al.’s (2008) and the reader can refer to their discussion regarding the comparison with experimental mea- surements.
In summary, our results show that the effect of pressure on Mg diffusion in forsterite is small, at least from the point of view of theoretical calcula- tions. The formation activation volume, ∆Vf, is of the order of 4.7 cm3/mol and the migration activation volume, ∆Vm, is between 0.3 and 0.7 (less than 1 cm3/mol). The P (∆Vf+ ∆Vm) term of the activation enthalpy of diffusion is only of the order of 50 kJ/mol at 10 GPa. Most of the pressure effect is on the formation of the defect, and very little on its migration. This is not surprising since Mgi at the saddle point is already experiencing a constriction.
Comparing our calculations with experimental results is difficult as our model is that of an unrealistically pure forsterite. It is usually assumed that the calcu- lated migration energy/volume corresponds to the extrinsic diffusion regime, and formation+migration to the intrinsic regime. This is of course a simplis- tic view of diffusion (see discussions in Chakraborty et al., 1994; Jaoul et al., 1995; Chakraborty, 1997) to which our model ties us. Experimental measure- ments of the activation volume of Mg diffusion in olivine (see a review in B´ejina et al., 2003) ranges from about 0 (Jaoul et al., 1995) to 7 cm3/mol (Holzapfel et al., 2007). Interestingly, for Mg self-diffusion in forsterite, this range is much narrower: Farver et al. (1994) gives ∆V ≤ 1 cm3/mol for grain- boundary diffusion and Chakraborty et al. (1994) estimated ∆V to be between
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1 and 3.5 cm3/mol for bulk diffusion along [001], both at T = 1100◦C. These experiments were done at relatively low temperature compared to the melt- ing temperature Tf, with T /Tf ≃ 0.6, and one can consider that they were conducted in an extrinsic regime. As discussed in the above references, in- trinsic diffusion is difficult to reach in silicates and has therefore rarely been measured. One example is Ca diffusion in synthetic diopside (Dimanov and Ingrin, 1995) with the intrinsic regime starting at T /Tf = 0.91. Considering uncertainties in both experiments and our calculations, and that our model ignores the eventual role of impurities, our value of ∆Vm agrees relatively well with laboratory measurements (at least with the lowest estimates).
Fe-Mg interdiffusion experiments performed at low temperature (600-900◦C) in San Carlos olivine along [010] (Jaoul et al., 1995) were also probably in the extrinsic regime. These authors measured an activation volume of 1.0 ± 0.9 cm3/mol (corrected for pO2), comparable to our value of 0.3 cm3/mol for migration. Note that this could just be fortuitous and calculations of the activation volume along [010] are in progress.
In other cases (Misener, 1974; Farber et al., 1994, 2000; Chakraborty, 1997;
Holzapfel et al., 2007), activation volumes for divalent cation diffusion in Fe- bearing olivines were measured between < 3.5 (Chakraborty et al., 1999) and 7 cm3/mol (Holzapfel et al., 2007). These are more difficult to compare with our calculations, and are neither intrinsic nor extrinsic but fall within an intermediate regime (the “constrained extrinsic” regime of Jaoul et al.
(1995) or “transition metal-extrinsic diffusion” of Chakraborty (1997)). It is nevertheless surprising that in this intermediate diffusion regime, pressure has the same or a bigger effect than for our purely intrinsic regime. If we assume that, in this regime, migration is also only slightly affected by pressure, then
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the vacancy population linked to impurities (as defined by Chakraborty, 1997) diminishes notably with increasing pressure. Another possibility is the effect of protons on diffusion. The activation volume of Fe-Mg diffusion in olivine in a water-satured environment was measured to be 16 ± 6 cm3/mol (Hier- Majumder et al., 2005). Therefore water has a strong effect on ∆V and the high values of ∆V could very well be linked to this effect. Also, the most popular calibration used to estimate the OH amount in anhydrous minerals (Paterson, 1982) has recently been modified (Bell et al., 2003; Koga et al., 2003). Previous OH concentration estimates should be multiplied by at least 3, a correction that applies to most diffusion experiments, including the recent one by Holzapfel et al. (2007).
In conclusion, divalent cation diffusion in olivine has been studied for more than 50 years and is now fairly well understood. With the combination of both experiments and theoretical calculations, the detailed mechanisms and influence of intensive and extensive parameters are better known. But, in the field of diffusion at high-pressure, a few discrepancies need to be addressed.
Here, we tried to decipher the role of pressure on Mg diffusion in forsterite, starting with the simplest model. We have shown, by various methods, that pressure has considerable influence on the formation of defects in forsterite, but only minimal impact on their migration. The activation volume can be considered independent of pressure over the entire upper mantle. This work, together with Walker et al.’s (2008), are the first steps of a theoretical study that will explore the role of Fe, OH and impurity contents, as well as different diffusion mechanisms and paths.
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Acknowledgements
We would like to add a special thought here for our friend, Olivier Jaoul. We thank two anonymous reviewers and our guest editor who helped to improve the manuscript.
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Fig. 1. Principle of the “manual” determination of the Mg diffusion path. At different positions along the ~c axis (direction of diffusion), a map of the energy of Mgiplaced at various coordinates on the [001] plane is determined. The diffusing Mg follows the path (arrows) defined by the energy minimum of each surface.
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Fig. 2. Formation energies of Mg vacancies (top graph: circles are for the M1 vacancy and squares for M2) and interstitial (lower graph) as a function of pressure. Black symbols are for energies calculated with the Mott-Littleton approach and grey with the supercell.
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Fig. 3. Energy of VMg diffusion along the ~c axis of forsterite as a function of pres- sure. Top graph: total diffusion energy; middle graph: formation energy of a Mg Frenkel defect; migration energy of VMg. Black symbols are for results using the Mott-Littleton approach, grey using the supercell and open diamonds for ab initio (migration only).
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Fig. 4. Schematic views of the forsterite structure in the [010] (top) and [100] (bot- tom) directions. Two VMg were placed in the M1 sites and the diffusion path is shown by the black line. The white dashed line represents the (001) mirror plane.
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Fig. 5. Energy vs. distance of VMg migration along the ~c axis of forsterite obtained with ML. The bottom axis represents the distance (fractional coordinates) along ~c between two Mg(M1) vacancies. Black symbols are for results at 0 GPa and grey symbols at 10 GPa. The open diamonds are results of calculations using the RFO routine at various pressures (from bottom to top: 0, 2, 5, 7.5 and 10 GPa). The dashed vertical line represents the (001) mirror plane.
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Table 1
Parameters for the set of potentials, THB1.
Charges
Ions Core Shell Core-shell spring constant (eV/˚A2)
Mg 2.0
Si 4.0
O2− 0.848190 -2.848190 74.92038
Buckingham potential
A (eV) ρ (˚A) C (eV.˚A6)
Si–O2− 1283.90734 0.32052 10.66158
O2−–O2− 22764.0 0.149 27.88
Mg–O2− 1428.5 0.29453 0.0
Harmonic three-body potential
k (eV/rad−2) θ0 (degrees)
O2−–Si–O2− 2.09724 109.47
The Buckingham potential is written, ϕ(rij) = A0exp (−rij/ρij) − C0/rij6, where
A0, Cij and ρ0 are fitted parameters describing the interaction of two species, i and j,
separated by a distance rij, separated by a distance rij, and ϕ(rij) is the contribution
to the total energy from this interaction. A cut-off of 12 ˚A was used for the Buckingham potential.
The harmonic three-body term is expressed as 12k(θ − θ0), θ being the obtuse angle of O-Si-O.
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Table 2
Formation energies (in eV) of various point defects in Fo at 0 GPa. W&C94: Wright and Catlow (1994); J95: Jaoul et al. (1995); RB00: Richmond and Brodholt (2000);
W07: Walker et al. (2008).
Defects This work W&C94 J95 RC00 W07
ML Supercell
VMg(M1) 24.47 24.43 24.47 24.5 24.43 24.48
VMg(M2) 26.39 26.24 26.4 26.25 26.40
Mgi −17.76 −17.93 −17.8 −17.75
VSi′′′′ 100.70 101.76 100.81
VO1•• 27.99 28.0 27.96 27.97
VO2•• 24.85 25.23 25.2 24.35 25.20
VO3•• 24.46 24.4 24.32 24.54
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Table 3
Formation energies (in eV) of Mg-related point defects in Fo as a function of pres- sure. J95: (Jaoul et al., 1995); RC00: (Richmond and Brodholt, 2000).
Defects This work J95 RC00
0 GPa 10 GPa 10 GPa 15 GPa
ML Supercell ML Supercell
VMg(M1) 24.47 24.43 24.71 24.60 24.7 24.71
VMg(M2) 26.39 26.24 26.91 26.65 26.9 26.84
Mgi −17.76 −17.93 −17.09 −17.32 −17.1
